Convective shells in the interior of Cepheid variable stars: overshooting models based on hydrodynamic simulations

Arthur Le Saux; Dimitar Vlaykov; Isabelle Baraffe; Jane Pratt; Joyce Ann Guzik; Mary-Geer Dethero; Maxime Stuck; Tom Goffrey

REVIEW 4 major objections 4 minor 134 references

Convective shells in the interior of Cepheid variable stars: overshooting models based on hydrodynamic simulations

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that in Cepheid interiors, overshooting above the inner convective shell reaches all the way to the outer envelope, while the ratio of lower to upper overshoot lengths drops as the shell deepens and thickens.

desk verdict A genuinely new statistical decomposition of overshoot plumes, but the paper's central ratio trend is undermined by right-censoring at the outer simulation boundary and needs a censoring-aware refit. read the letter →

arxiv 2505.04900 v1 pith:JH2QTQAS submitted 2025-05-08 astro-ph.SR

Maxime Stuck , Jane Pratt , Isabelle Baraffe , Joyce Ann Guzik , Mary-Geer Dethero , Dimitar Vlaykov , Tom Goffrey , Arthur Le Saux This is my paper · ORCID

classification astro-ph.SR

keywords convectiveovershootingshellsCepheidvariableshydrodynamicsimulationsextremevaluetheoryGammamixturemodelboundarymixingsuper-mixinglayer

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Cepheid variable stars serve as cosmic distance markers, yet their evolution models disagree with pulsation-based masses by 10–20%. The paper argues that one missing ingredient is a realistic treatment of the inner convective shell that appears in the Z-bump opacity region of intermediate-mass Cepheids. Using 2D hydrodynamic simulations of six stars from 5.75 to 9 solar masses, it shows that overshooting plumes above this shell travel far enough to fill the entire radiative layer up to the outer convective envelope. To quantify the mixing, the paper fits the distribution of plume penetration distances as a mixture of two Gamma distributions, one for frequent weak overshooting and one for rare strong overshooting; the mean of the strong component defines a characteristic overshooting length on each side of the shell. The ratio of lower to upper strong-overshooting lengths stays below one and decreases as the shell becomes deeper and thicker, leading to a new diffusion coefficient, $D_{\rm shell}$ (Eq. 18), and to the idea of a 'super-mixing layer' where shell and envelope overshoot overlap.

What carries the argument

The central object is the mixture-of-two-Gamma model fitted to the PDF of the plume penetration distance $\Delta r_o = |r_o - R_{\rm CB}|/H_{p,\rm CB}$ (Eqs. 12–16). The weak Gamma component captures the 85–95% of plumes that stop just past the convective boundary; the strong Gamma component captures the 5–15% of rare, deep plumes, and its mean $\ell_{\rm so}$ becomes the characteristic overshooting length. The same fitted CDF feeds the new diffusion coefficient $D_{\rm shell}$ (Eq. 18), which keeps the whole strong-overshooting distribution instead of only the deepest plume used by the extreme-value-theory approach. The Weibull/GEVD analysis (Eq. 11) remains as a consistency check and provides the maximal overshoot length.

What would settle it

Re-run the same six stellar structures with the simulation domain extended through the outer convective envelope, or with a large radiative buffer above the shell, and rebuild the PDF of upper penetration distances; if the mean of the strong-overshooting Gamma component, the 90th-percentile length, or the ratio $\ell^L_{\rm so}/\ell^U_{\rm so}$ shifts by more than the quoted 1$\sigma$ errors, the truncated boundary is distorting the central result.

Watch

Extended reading notes

Core claim

The paper's central claim is that convective boundary mixing around a Cepheid's inner shell is not one process but two: frequent shallow plumes that barely cross the Schwarzschild boundary, and rare deep plumes that dominate mixing in the radiative zone. By fitting the probability density function of the dimensionless penetration distance $\Delta r_o$ with a mixture of two Gamma distributions, the analysis isolates the strong-overshooting population; its mean, $\ell_{\rm so}$, is proposed as the characteristic overshooting length on each side of the shell, while its 90th percentile matches the maximal length from extreme-value theory. Across the six simulations, the ratio $\ell^L_{\rm so}/\ell^U_{\rm so}$ is close to unity for shallow, thin shells and drops steadily as the shell lies deeper and becomes wider, and it is always below one. The paper further claims that for stars above about 6 solar masses the upper overshooting layer reaches through the whole radiative zone, and that when shell and envelope overshooting layers overlap the diffusion coefficient can exceed the local mixing-length value, so the two convective layers may behave as one super-mixing layer. From the mixture-model CDF, it proposes the diffusion coefficient $D_{\rm shell}$ (Eq. 18) as a calibrated input for 1D stellar evolution models.

Load-bearing premise

The results assume that truncating each simulation just below the outer convective envelope does not change how far overshooting plumes would really travel above the shell, even though for stars above 6 solar masses the strongest plumes regularly hit this artificial outer boundary and the measured upper overshooting distribution is therefore cut off at the edge of the domain.

Editorial extensions

If this is right

1D Cepheid evolution models can adopt $D_{\rm shell}$ (Eq. 18) with the parameter ranges of Eq. (19), replacing an uncalibrated overshooting length with a prescription derived from plume statistics on both sides of the shell.
For stars above about 6 $M_\odot$, the upper overshooting layer occupies the entire radiative zone between the shell and the outer envelope, so evolution models that leave that zone unmixed underestimate the chemical communication between the two convective layers.
The lower-to-upper overshoot ratio is not a single constant: the trend with shell depth and width explains why massive-star shells and thin A-type shells had previously yielded ratios of about 0.2 and 0.5.
Overlapping overshoot layers from the shell and the envelope can produce local diffusion larger than the mixing-length value, so the two adjacent convective zones may effectively merge into a super-mixing layer.
Using the 90th percentile of the strong-overshooting Gamma, at most about 1.5% of all overshooting plumes can reach the outer convective envelope directly, quantifying how much mass exchange crosses the radiative zone.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the trend in Figs. 12a,b continues beyond the simulated range, then for thicker and deeper shells the ratio $\ell^L_{\rm so}/\ell^U_{\rm so}$ would approach the factor of about 0.2 found in simulations of massive stars, unifying the previously scattered literature values; this extrapolation is not made in the paper.
Because the simulations omit radial pulsations, a testable extension is to impose the Cepheid pulsation on the shell boundaries and remeasure the two Gamma components; the characteristic lengths might then become phase-dependent, which would time-modulate $D_{\rm shell}$.
The mixture-model split into weak and strong overshooting could be applied to other convective boundaries, such as cores and envelopes, giving a common two-population language for overshooting across stellar regimes; the paper only demonstrates it for shells.
A 3D simulation of the same shells at comparable radial resolution would show whether the convection-roll geometry that dominates these thin shells is a 2D artefact; if the strong-overshooting Gamma mean changes, the absolute values of $\ell_{\rm so}$ but not necessarily the ratio trend would need revision.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing to know: the Gamma-mixture decomposition is a real methodological step, and the paper is honest about its own limits. Second: the stress-test concern is correct. The upper overshooting statistics are right-censored by a non-penetrative outer boundary, and the mixture fit does not account for it. That puts the headline trend (ℓL_so/ℓU_so decreasing with shell depth/thickness) on shaky ground, and it also contaminates the proposed diffusion coefficient.

What the paper does well: long 2D MUSIC simulations of six Cepheid-like structures, high radial resolution at the convective boundaries, more than 25 convective turnover times per run. The diagnostics (filling factor, inflow width, vertical kinetic flux) are clearly presented. The finding that overshoot above the shell reaches the outer boundary is a real result, even if it complicates the statistics. The mixture model is a sensible, if standard, way to split a skewed positive PDF into a frequent weak component and a rare strong component; I think the Gamma choice is well motivated by maximum entropy. The authors also explicitly flag that the six-model sample is too small to separate depth and width effects, and that the super-mixing layer needs confirmation. The self-citations to the group's earlier EVT work are appropriate rather than excessive.

The soft spots are the load-bearing ones. Section 3.3.1 fits the CDF with least squares and no censoring term. Plumes that would have gone past Ro are recorded at Ro. The SO component is exactly the tail being truncated, so its mean, 90th percentile, and weight are biased downward on the upper side. The bias is probably worst for ceph8 and ceph9, where the EVT already returns δ-like distributions at the boundary. The apparent drop of the ratio with depth/width could therefore come from domain geometry rather than stellar structure. The same censoring feeds Eq. 18 through Eq. 19, so the proposed mixing prescription inherits the bias. The authors acknowledge the boundary issue for EVT, but they do not carry it into the mixture fit. A censoring-aware likelihood, or one model with a larger outer radius, would resolve the question.

Minor point: the six points in Fig. 12 have correlated depth and width and no significance test; the authors say this themselves. Also, D_shell is a calibrated fit to the same runs, so it should be presented as calibration, not prediction; the authors mostly do this. The boundary issue is the one that needs fixing before the ratio can be trusted.

Who should read it: anyone working on convective boundary mixing in shell convection, and developers of 1D overshoot prescriptions. I would cite the mixture-model approach, but not the ratio trend as it stands. Verdict: this deserves a serious referee, not a desk reject. The referee should request a censoring-aware refit and a clear statement of how much of the trend survives.

Referee Report

4 major / 4 minor

Summary. This paper presents 2D MUSIC simulations of six 5.75–9 M_sun Cepheid models that contain an interior convective shell and an outer convective envelope, with the simulation domain truncated just below the envelope. The authors characterize the shell convection through filling factors and plume widths, then analyze convective boundary mixing using extreme value theory and a newly proposed mixture of two Gamma distributions fitted to the PDF of plume penetration distances. They report that strong overshooting above the shell fills the radiative layer, that the ratio of lower to upper strong-overshooting lengths decreases with shell depth and width (Figs. 12a,b), and they propose a diffusion coefficient for 1D stellar evolution models (Eq. 18), including the speculative idea of a 'super-mixing layer' formed by overlapping overshooting from the shell and the envelope.

Significance. The paper addresses an important gap: multidimensional simulations of the interior convective shells of Cepheids are rare, and 1D overshooting prescriptions for such shells are poorly calibrated. The simulations are long (25-138 convective turnover times) and well resolved (more than 100 grid cells per pressure scale height at the convective boundaries), and the mixture-model decomposition of overshooting PDFs is a genuinely new statistical tool that could provide physically motivated overshooting lengths and a smooth diffusion profile. If the censoring and statistical-significance issues are resolved, the proposed relationship between shell geometry and overshooting asymmetry would be directly useful for stellar evolution modelling, and the super-mixing-layer hypothesis is stimulating even though its quantitative form relies on assumed envelope parameters.

major comments (4)

[Sec. 3.2.2 and Sec. 3.3.1, Table 3, Fig. 12] The upper-side overshooting statistics are right-censored by the artificial outer boundary. For all models except ceph5, plumes frequently reach the outer radius Ro, and because the radial boundaries are non-penetrative (Sec. 2) the vertical kinetic energy flux vanishes there; any plume that would have penetrated farther is recorded at Delta_ro_max = (Ro - R_U_CB)/Hp_U_CB. The mixture-model fit in Sec. 3.3.1 minimizes least squares on the CDF and contains no censoring term, so the SO component of the upper PDF, which is precisely the tail affected by the cutoff, has biased values of the mean ell_U_so, the 90th percentile, and the mixture weight. The bias is expected to be largest for ceph8 and ceph9, where the EVT already gives delta-like distributions at the boundary. Consequently the decrease of ell_L_so/ell_U_so with depth and width shown in Fig. 12 could be partly a domain-size artifact rather than a physical trend. Please provide a censoring-aware refit (e.g., maximum likelihood with a survival contribution for Delta_ro = Delta_ro_max) or repeat with a larger outer boundary, and quantify how ell_U_so and the ratios in Fig. 12 change.
[Sec. 3.3.2, Fig. 12] The central trend is inferred from only six models in which shell depth (R_L_CB/R) and shell width ((R_U_CB - R_L_CB)/Hp) are correlated with each other. The error bars in Fig. 12 are the standard deviations of the fitted SO Gamma distributions, not the uncertainties of the means or of the fit parameters, and no significance test, correlation coefficient, or confidence interval is reported for the linear regression line. Given the scatter and the overlap of the error bars, the statement that the ratio 'depends directly' on depth and width is not statistically established. The authors' caveat at the end of Sec. 3.3.2 acknowledges this, but the abstract and conclusions present the trend as a result. Please add significance testing (e.g., Spearman or Pearson correlation with bootstrap or a hierarchical model) and report parameter uncertainties rather than distribution widths.
[Sec. 3.3.3, Eqs. (18) and (21)] The proposed diffusion coefficient is dimensionally inconsistent as written. The variable Delta_ro in Eq. (12) is normalized by Hp,CB, and the mixture parameters in Eqs. (19) are quoted in units of Hp,CB, but the arguments of the incomplete Gamma functions in Eq. (18) contain (R_CB - r)/R, i.e., a stellar-radius normalization. As a result, D_shell is not the CDF-based quantity described in the text, and Fig. 13 cannot be reproduced from Eq. (18). Please replace the /R factors with /Hp,CB (or otherwise consistently define the dimensionless radial coordinate in the Gamma CDF) and regenerate the figure. The same issue affects Eq. (21).
[Sec. 3.3.3, Fig. 13] The super-mixing layer and the full-star diffusion coefficient D_MM are constructed by assuming the envelope overshooting parameters are equal to the lower-shell parameters (mu_E = mu_L, alpha_E = alpha_L, lambda_E = lambda_L) and that D_envelope_MLT = 0.8 D_MLT, with no simulation containing the envelope. The text does state these are assumptions, but the conclusion that a super-mixing layer with D/D_MLT > 1 could exist is an extrapolation, not a simulation result. Please mark this explicitly in the figure caption and in Sec. 4 so that readers do not mistake the assumed envelope model for a calibrated prediction.

minor comments (4)

[Abstract and Sec. 3.2.2] The claim that overshooting above the convective shell 'fills the space between these convectively unstable layers' should be qualified: for M > 6 M_sun this is inferred from plumes that hit the outer boundary of a domain truncated just below the envelope, so the filled region is the simulated gap, not an independently converged overshooting length.
[Table 3 caption] The caption describes the standard deviation of the SO Gamma distribution as an 'error on the mean'; it is the distribution width, not a standard error of the mean. Please reword to avoid implying a smaller statistical uncertainty.
[Fig. 12] The linear regression line is drawn without reporting its equation, R-squared value, or uncertainty; either report these quantities or remove the line to avoid implying a quantified fit.
[Sec. 3.3.1] The mixture model has seven free parameters fitted by nonlinear least squares, but no goodness-of-fit statistic or parameter covariance is reported; adding at least a residual metric and a bootstrap uncertainty on ell_so would strengthen the comparison between models.

Circularity Check

2 steps flagged · score 6.0 of 10

Upper-overshoot 'fills the radiative zone' result is the truncated boundary by construction, and the mixture-model tail fit inherits that censoring.

self definitional [Section 3.2.2, Eqs. (6), (9), (10)]
"Because sometimes plumes reach the outer simulation boudary, maximal-in-time plumes often reach the simulation boundary. Therefore, the EVT approach produces an overshooting length equal to the depth of the radiative layer for all of the stars we study except for ceph5."

The upper maximal overshooting length is defined as rmax(t) = max_theta ro(theta,t) (Eq. 9), where ro is the first zero of the vertical kinetic energy flux Fk (Eq. 6). With non-penetrative radial boundaries (Section 2), Fk = 0 at the outer radius Ro, so any plume that would have penetrated farther is recorded at Ro. The 'EVT result' of an overshooting length equal to the depth of the radiative layer is therefore an identity with the placement of the simulation boundary, not an independent measurement of stellar overshooting. The paper explicitly acknowledges this for all models except ceph5, yet the abstract retains 'overshooting above the convective shell fills the space between these convectively unstable layers' as a finding.
fitted input called prediction [Section 3.3.2, Eqs. (12)-(16), Table 3]
"The 90% SO length scale reinforces the idea that the upper overshooting layer fills its radiative zone; it also provides a picture of how many plumes reach the outer convective envelope."

The upper SO distribution is fitted to the same PDF(Delta_ro) that is right-censored at the outer boundary: plumes that would go beyond Ro are recorded at Ro, so the fitted Gamma tail parameters, the SO mean, and the 90th percentile are all constrained by the artificial cap. Comparing this 90th percentile with the EVT length and concluding that it 'reinforces' the filling of the radiative zone is the same boundary effect restated in a new distribution. The paper does not apply a censoring correction in the Levenberg-Marquardt CDF fit described in Section 3.3.1, so the upper-side 'confirmation' reduces by construction to the domain truncation.

full rationale

The main derivation chain is an empirical calibration: MUSIC simulations produce PDFs of plume penetration, a two-Gamma mixture is fitted to those PDFs, and the fitted SO parameters define characteristic lengths and a proposed diffusion coefficient D_shell (Eq. 18). This is not circular in the sense of the diffusion coefficient being a fitted prescription rather than an independent prediction; the paper does not claim to have tested D_shell against external data. The ratio trend in Figs. 12a,b is a regression on the fitted mixture quantities, which is an ordinary data summary rather than a derivation of the inputs. The genuinely load-bearing circular/artifactual element is the upper boundary: for all models except ceph5, the EVT maximal overshoot length equals the distance to the outer simulation boundary by construction (Section 3.2.2), and the mixture-model upper-tail parameters are fitted to that same right-censored sample. Therefore the headline result that overshooting fills the radiative zone, and the upper-side quantities used in the ratio and in D_shell, inherit the domain size rather than being independent stellar-physics measurements. The paper is candid about the EVT saturation but does not propagate the censoring into the mixture-model estimates or the ratio trend. No load-bearing self-citation chain appears: the cited Pratt et al. (2017) EVT and diffusion-coefficient framework is a reusable external method, not a uniqueness theorem invoked to forbid alternatives, so it does not by itself raise the circularity score.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The central statistical results are loaded with fitted parameters: seven mixture parameters per PDF, assumed envelope parameters for the super-mixing layer, and regression lines on the ratio. The main assumptions are the Schwarzschild boundary criterion, the kinetic-flux definition of plume extent, 2D adequacy, homogeneous composition, and, most importantly, that the outer boundary truncation does not bias the overshooting statistics.

free parameters (2)

Gamma mixture parameters (WO and SO: location mu, shape alpha, scale lambda; weight w) = Per simulation; SO ranges in Eq. (19): mu_U_so ~ 0.5-0.9 H_U_p,CB, mu_L_so ~ 0.7-1.0 H_L_p,CB, alpha_U_so ~ 2.0-5.0…
Seven free parameters per overshooting layer are fitted to the simulated PDF of penetration distances via Levenberg-Marquardt; the resulting SO distribution defines the overshooting length and the diffusion coefficient in Eq. (18). These are calibrated to the same simulations, not predicted.
Envelope overshooting parameters (mu_E, alpha_E, lambda_E, D_envelope_MLT) = mu_E_so = mu_L_so, alpha_E_so = alpha_L_so, lambda_E_so = lambda_L_so; D_envelope_MLT = 0.8 D_MLT
Assumed by hand in Section 3.3.3 to illustrate the super-mixing layer; not derived or measured, because the simulations do not include the outer convective envelope.

assumptions (6)

domain assumption The Schwarzschild criterion defines the convective boundaries (nabla_ad = nabla_rad) and the unstable layers.
Used throughout to identify the convective shell and envelope boundaries in the MESA models and simulations (Section 2, Fig. 2).
domain assumption The first zero of the vertical kinetic energy flux marks the radial extent of an overshooting plume.
This diagnostic, from Hurlburt et al. and Pratt et al., is used to compute the plume extent and define the PDF on which all statistical results rest (Section 3.1, Eq. 6).
domain assumption 2D simulations adequately reproduce the overshooting lengths relevant here despite higher velocities in 2D.
The paper argues that radial velocities at convective boundaries and overshooting lengths are similar in 2D and 3D based on Pratt et al. (2020) and Dethero et al. (2024), but the central results are obtained with 2D only (Section 1, Table 2).
domain assumption The simulation domain truncation just below the outer convective envelope does not invalidate the overshooting statistics.
Plumes overshooting above the shell frequently reach this artificial outer boundary (Section 3.2.2), and the analysis does not correct for censoring, so the statistical decomposition implicitly assumes boundary hits are equivalent to natural stopping distances.
standard math The Gamma distribution is the maximum-entropy model for a strictly positive random variable, motivating the mixture.
The mixture model rests on the entropy argument from Park & Bera (2009), quoted in Section 3.3.1.
domain assumption Homogeneous chemical composition is assumed in the hydrodynamic simulations.
Stated in Section 2; ignores composition gradients which could affect convective boundary mixing.

invented entities (1)

Super-mixing layer
purpose: A layer between the inner convective shell and the outer convective envelope where overshooting from both zones overlaps, producing more efficient mixing than in either convection zone and potentially merging them.
Introduced in the abstract and Section 4 based on the extrapolated diffusion coefficient D_MM (Eq. 20) with assumed envelope parameters; no simulation including both convective layers confirms its existence, as the authors state.

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Cite this review

Pith. "Pith review of Convective shells in the interior of Cepheid variable stars: overshooting models based on hydrodynamic simulations." pith.science (2026). https://pith.science/paper/JH2QTQAS

@misc{pith2026250504900,
  author       = {Pith},
  title        = {Pith review of: Convective shells in the interior of Cepheid variable stars: overshooting models based on hydrodynamic simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JH2QTQAS}},
  note         = {Machine review of arXiv:2505.04900}
}

read the original abstract

Because Cepheid variable stars have long been used as a cosmic benchmark, the accuracy of stellar evolution models for Cepheids have wide-reaching effects. Our goal is to provide a detailed multi-dimensional picture of hydrodynamic convection and convective boundary mixing in the interior of Cepheids. We perform 2D hydrodynamic simulations of six stars with the fully compressible Multidimensional Stellar Implicit Code (MUSIC). Our simulations do not model the radial pulsations but focus on the interior structure of Cepheids, which involves an interior convective shell and a convective envelope. We develop a new statistical analysis to examine overshooting in this inner convection zone. Using the extreme value theory, we find that overshooting above the convective shell fills the space between these convectively unstable layers. We develop a new statistical analysis that provides a clearer picture of how overshooting fills this layer, and also allows us to formulate a detailed comparison between overshooting above and below the convective shell. Our analysis effectively decomposes the overshooting layer into two layers: a weak and a strong overshooting layer. Statistically, this is accomplished by decomposing the strongly non-Gaussian probability density function into a mixture of Gamma distributions. Using our mixture model, we show that the ratio of overshooting lengths above and below the convective shell depends directly on the radial extent of the convective shell as well as its depth in the star. We propose a new form for the diffusion coefficient, which addresses the need for overlapping overshooting layers between convective shells. We introduce the idea of super-mixing layer where overshooting from both the convective shell and the convective envelope results in efficient mixing and could be viewed as merging the two adjacent convective zones.

Figures

Figures reproduced from arXiv: 2505.04900 by the authors.

**Figure 1.** Schematic of the typical structure of the Cepheids we study in this work with MUSIC. Convective regions are colored, including: a tiny convective core, an inner convective shell surrounded by radiative zones, and a thin outer convective envelope. The simulation domain is a spherical shell, and is outlined with black; it encapsulates the interior convection zone along with the surrounding radiative zones, and is trun… view at source ↗

**Figure 2.** (a) HR diagram and (b) radial profile of the Schwarzschild discriminant, for the six stars we simulate with MUSIC. Symbols indicate the Cepheids we are simulating and the pink shaded area indicates the instability strip for the evolutionary tracks. Shaded regions also highlight the internal convection zones around which our hydrodynamic simulations are centered. azimuthal symmetry. In the table, the inner and oute… view at source ↗

**Figure 3.** Visualizations of (from left to right) vorticity magnitude, radial velocity and angular velocity of the 8M⊙ simulation after 30τconv of steadystate convection. The zero point for vorticity is in dark red in the left panel. In the center and right panels, outwards flows are in red while inwards flows are in yellow; the zero point in velocity is black. The radial velocity ranges from −9.25 × 105 cm/s to 9.25 × 105 cm… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Radial profile of the time-averaged RMS velocity scaled by Lstar/10351/3 for a range of stellar masses. Vertical lines indicate the Schwarzschild convective boundaries for each star. element V. We use the convective turnover time to verify that convection has reache…

**Figure 5.** Figure 5: Radial profile of the time-averaged volume percentage filling factor of the inward moving plumes σvp,in vs the star’s internal radius, in units of the total stellar radius R for the 6M⊙ simulation. The shaded area indicates one standard deviation above and below this a…

**Figure 6.** Figure 6: Width of inflowing plumes Win for the 5.75 and the 9M⊙ simulations as a function of the star’s radius. Shaded areas indicate one standard deviation above and below the time-averaged line. Thin vertical lines indicate the radial position of the Schwarzschild boundarie…

**Figure 7.** Figure 7: (a) Angular structure of the overshooting layer after 114τconv of steady-state convection in the simulation of the 6M⊙ Cepheid. The overshooting length in this illustration is determined by zeros of the vertical kinetic flux. The Schwarzschild convective boundaries bet…

**Figure 8.** Figure 8: Radial profiles of the convective flux for the 6M⊙ Cepheid. The shaded area indicates one standard deviation around the mean value. Thin vertical lines indicate the convective boundaries determined by the Schwarzschild criterion. Black dashed lines indicates one pressu…

**Figure 10.** Figure 10: Natural logarithm of the negative natural logarithm of the cumulative distribution functions of maximal overshooting length, ∆rmax for each simulation below the convective shell. Black lines indicate the best fit with the Weibull distribution, defined in eq. (11). to…

**Figure 12.** Figure 12: Ratio of the extent of the lower overshooting layer (ℓ L so) over the extent of the upper overshooting layer (ℓ U so) given by the mixture model, (a) vs. the depth of the convective shell, measured as R L CB/R, and (b) vs. the width of the convective shell measured as…

**Figure 13.** Figure 13: Diffusion coefficient for the 7M⊙ (a) Cepheid. Heavy black lines indicate the convective boundaries of both the convective shell and the outer convection zone. The thin black dotted line is the diffusion coefficient of the convective shell based on the mixture model …

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